Mathematics Grade 12 Paper 2 Challenge Paper 2026 Question Paper

Copyright reserved Please turn over Challenge hlayiso mock exams Practice Series: Grade 12 NSC Preparation WWW.HLAYISO.COM NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS P2 CHALLENGE PAPER 2026 MARKS: 150 TIME: 3 hours This question paper consists of 12 pages, 1 information sheet and a SPECIAL ANSWER BOOK.

Mathematics/P2 2 NSC Challenge Hlayiso/Challenge 2026 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 11 questions. 2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining your answers. 4. Answers only will NOT necessarily be awarded full marks. 5. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 6. If necessary, round off answers to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. An information sheet with formulae is included at the end of the question paper. 9. Write neatly and legibly. 10. This is a challenge paper: it is deliberately harder than the NSC examination. Copyright reserved Please turn over

Mathematics/P2 3 NSC Challenge Hlayiso/Challenge 2026 QUESTION 1 The travelling times (in minutes) of 11 learners from a school in Soweto were recorded one morning. The data, arranged in ascending order, is given below. x and y are whole numbers. 25 28 33 37 x 45 48 52 56 y 71 The mean travelling time is 46 minutes and y is 25 more than x. 1.1 Calculate the values of x and y. (3) 1.2 Draw a box and whisker diagram of the data on the number line provided in the ANSWER BOOK. (3) 1.3 Calculate the standard deviation of the travelling times. (2) 1.4 How many learners have a travelling time that lies within ONE standard deviation of the mean? (2) 1.5 Because of road works, every learner’s travelling time increases by 10%. A new taxi route then reduces every learner’s (increased) travelling time by 5 minutes. Write down the new mean and the new standard deviation of the travelling times. (2) [12] Copyright reserved Please turn over

Mathematics/P2 4 NSC Challenge Hlayiso/Challenge 2026 QUESTION 2 The owner of a tuck shop in Polokwane recorded the maximum temperature (x, in ◦C) and the number of cold drinks sold (y) on 8 school days. The data is shown in the table and on the scatter plot below. MAXIMUM TEMPERATURE (IN ◦C) (x) 18 21 23 25 26 28 30 33 NUMBER OF COLD DRINKS SOLD (y) 34 45 49 58 55 66 70 79 15 20 25 30 35 30 40 50 60 70 80 90 Maximum temperature (in ◦C) Number of cold drinks sold SCATTER PLOT 2.1 Determine the equation of the least squares regression line for the data. (3) 2.2 Write down the value of the correlation coefficient and describe the relationship between the temperature and the number of cold drinks sold. (2) 2.3 Predict the number of cold drinks that will be sold on a day when the maximum temperature is 27 ◦C. (2) 2.4 On a ninth day the maximum temperature was 35 ◦C. When the data for this day is included, the mean number of cold drinks sold per day increases by 3. Calculate the number of cold drinks sold on the ninth day. (3) [10] Copyright reserved Please turn over

Mathematics/P2 5 NSC Challenge Hlayiso/Challenge 2026 QUESTION 3 In the diagram, A(1 ; 6), B(−2 ; 0) and C(3 ; 0) are the vertices of △ABC. D is a point on AC such that BD ⊥AC. x y O A(1 ; 6) B(−2 ; 0) C(3 ; 0) D 3.1 Calculate the size of B ˆAC. (4) 3.2 Determine the equation of BD in the form y = mx+c. (3) 3.3 Determine the coordinates of D. (3) 3.4 Prove that △ABD is isosceles. (2) 3.5 Calculate the area of △BDC. (3) 3.6 P(x ; y) is a variable point above the x-axis which moves such that BˆPC = B ˆAC at all times. Determine the equation of the path along which P moves. Write your answer in the form (x−a)2 +(y−b)2 = r2. (4) [19] Copyright reserved Please turn over

Mathematics/P2 6 NSC Challenge Hlayiso/Challenge 2026 QUESTION 4 In the diagram, M is the centre of the circle with equation x2 +y2 −4x+6y−12 = 0. P(6 ; 0) is a point on the circle. The tangent to the circle at P cuts the y-axis at T. TQ is the second tangent from T to the circle, touching the circle at Q. x y O M P(6 ; 0) T Q 4.1 Determine the coordinates of M and the length of the radius. (3) 4.2 Determine the equation of the tangent TP in the form y = mx+c. (4) 4.3 Write down the coordinates of T and hence calculate the length of TP. (3) 4.4 Determine the coordinates of Q. (5) 4.5 Determine the equation of the circle that passes through M, P, T and Q. (3) 4.6 For which values of k will the line y = −4 3x + k cut the circle with centre M at two distinct points? (3) [21] Copyright reserved Please turn over

Mathematics/P2 7 NSC Challenge Hlayiso/Challenge 2026 QUESTION 5 5.1 Given: sin(A+B) = 4 5 and sin(A−B) = 1 5. Without using a calculator, determine the value of tanA tanB. (4) 5.2 Without using a calculator, simplify the following expression to a single number: cos(540◦−2x) sin(x−450◦).cos(−x) +tan(x−180◦).tan(540◦+x) (6) 5.3 Without using a calculator, determine the value of cos15◦−sin15◦. Leave your answer in simplest surd form. (3) [13] QUESTION 6 Given: f(x) = 1−cos2x+sin2x 1+cos2x+sin2x 6.1 Prove that f(x) = tanx for all values of x for which f(x) is defined. (4) 6.2 Determine the values of x in the interval x ∈[−180◦; 180◦] for which f(x) is undefined. (3) 6.3 Hence, determine the general solution of the equation [ f(x)]2 = 2 f(x)+3. (6) [13] Copyright reserved Please turn over

Mathematics/P2 8 NSC Challenge Hlayiso/Challenge 2026 QUESTION 7 In the diagram, the graphs of f(x) = asinbx and g(x) = cos(x + p) are drawn for the interval x ∈[−180◦; 180◦], where a > 0, b > 0 and −180◦< p < 180◦. The point (45◦; 1) lies on f and (30◦; 1) is a turning point of g. x y 0 (45◦; 1) (30◦; 1) f g −180◦ −150◦ −120◦ −90◦ −60◦ −30◦ 30◦ 60◦ 90◦ 120◦ 150◦ 180◦ −1 −0,5 0,5 1 7.1 Write down the values of a, b and p. (3) 7.2 Determine, without using a calculator, the values of x in the interval x ∈ [−180◦; 180◦] for which f(x) = g(x). (4) 7.3 For which values of x in the interval x ∈[0◦; 180◦] is f(x) ≥g(x)? (3) 7.4 Describe a horizontal translation of the graph of g that will produce the graph of y = −sinx. (2) [12] Copyright reserved Please turn over

Mathematics/P2 9 NSC Challenge Hlayiso/Challenge 2026 QUESTION 8 In the diagram, PQ is a vertical tower of height h metres. Q, R and S are points in the same horizontal plane. The angles of elevation of P from R and from S are both equal to α. R ˆQS = 2θ and RS = d metres. M is the midpoint of RS. α α 2θ R S Q P M h d 2 d 2 8.1 Show that h = d tanα 2sinθ (5) 8.2 Calculate the height of the tower if d = 80 m, α = 32◦and θ = 50◦. (2) 8.3 Using the values in QUESTION 8.2, calculate the angle of elevation of P from M. (3) [10] Copyright reserved Please turn over

Mathematics/P2 10 NSC Challenge Hlayiso/Challenge 2026 Give reasons for your statements in QUESTIONS 9, 10 and 11. QUESTION 9 9.1 In the diagram, O is the centre of the circle and A, B and C are points on the circle. TAS is a tangent to the circle at A. Chords AB, BC and AC are drawn. O A B C T S Use the diagram to prove the theorem which states that the angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle in the alternate segment, that is, prove that B ˆAS = A ˆCB. (6) 9.2 In the diagram, O is the centre of the circle and ABCD is a cyclic quadrilateral. PA is a tangent to the circle at A. AD = DC and P ˆAD = 35◦. BD, AC, OA and OC are drawn. O 35◦ A B C D P Calculate, with reasons, the size of: 9.2.1 A ˆBD (2) 9.2.2 A ˆBC (2) 9.2.3 O ˆAC (3) [13] Copyright reserved Please turn over

Mathematics/P2 11 NSC Challenge Hlayiso/Challenge 2026 QUESTION 10 In the diagram, D is a point on side BC of △ABC such that BD : DC = 3 : 2. E is the midpoint of AD. BE is produced to meet AC at F. A B C D E F 10.1 Determine, with reasons, the ratio AF : FC. (5) 10.2 Determine, with reasons, the ratio BE : EF. (4) 10.3 Calculate the value of area of △AEF area of △ABC (4) [13] Copyright reserved Please turn over

Mathematics/P2 12 NSC Challenge Hlayiso/Challenge 2026 QUESTION 11 In the diagram, O is the centre of the circle and AB is a diameter. AB is produced to D. DC is a tangent to the circle at C. DC is produced to E such that AE ⊥DE. AC, BC and OC are drawn. Let B ˆCD = x. x O A B D C E 11.1 Name, with reasons, TWO other angles that are each equal to x. (4) 11.2 Prove that AC bisects E ˆAB. (3) 11.3 Prove that △DCB ||| △DAC. (3) 11.4 If DB = 4 units and DC = 8 units, calculate the length of AE. (4) [14] TOTAL: 150 Copyright reserved Please turn over

Mathematics/P2 13 NSC Challenge Hlayiso/Challenge 2026 INFORMATION SHEET x = −b± √ b2 −4ac 2a A = P(1+ni) A = P(1−ni) A = P(1−i)n A = P(1+i)n Tn = a+(n−1)d Sn = n 2 [ 2a+(n−1)d ] Tn = arn−1 Sn = a(rn −1) r −1 ; r̸ = 1 S∞= a 1−r ; −1 < r < 1 F = x[(1+i)n −1] i P = x[1−(1+i)−n] i f ′(x) = lim h→0 f(x+h)−f(x) h d = √ (x2 −x1)2 +(y2 −y1)2 M (x1 +x2 2 ; y1 +y2 2 ) y = mx+c y−y1 = m(x−x1) m = y2 −y1 x2 −x1 m = tanθ (x−a)2 +(y−b)2 = r2 In △ABC : a sinA = b sinB = c sinC a2 = b2 +c2 −2bc.cosA area △ABC = 1 2ab.sinC sin(α +β) = sinα.cosβ +cosα.sinβ sin(α −β) = sinα.cosβ −cosα.sinβ cos(α +β) = cosα.cosβ −sinα.sinβ cos(α −β) = cosα.cosβ +sinα.sinβ cos2α =      cos2 α −sin2 α 1−2sin2 α 2cos2 α −1 sin2α = 2sinα.cosα ¯x = ∑x n σ2 = ∑n i=1 (xi −¯x)2 n P(A) = n(A) n(S) P(A or B) = P(A)+P(B)−P(A and B) ˆy = a+bx b = ∑(x−¯x)(y−¯y) ∑(x−¯x)2 Copyright reserved